

First place to start, get a video started playing in your browser, then CTRL+SHIFT+C (at least in Firefox+derivs) lets you highlight elements on the page and it will take you right to the spot in the source where it is. Might get lucky and find a mp4 link sitting right there for the taking.
Something tells me they have it a little more locked down, though. Might be able to monitor the network tab in Dev tools and filter for media to find the stream, which could also come through in chunks instead of one big file. Might even be obfuscated depending on how much money they put into the web side of things.
Barring all that, use OBS to screencap in realtime
Like a really shitty, dumb version of The Last Question. The prompt goes in now, AI brings it to fruition centuries later and we become the infirm people from WALL-E instead of transcending